For which L(s) will be these vectors linearly dependent?

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Discussion Overview

The discussion revolves around determining the values of L for which the vectors a, b, and c are linearly dependent. It also explores how the parameter ß relates to L when considering the vector v in the span of a, b, and c. The scope includes mathematical reasoning and conceptual clarification regarding linear dependence.

Discussion Character

  • Mathematical reasoning, Conceptual clarification, Debate/contested

Main Points Raised

  • One participant asks how to calculate L to make the vectors linearly dependent and how ß depends on L.
  • Another participant suggests using the definition of linear dependence, indicating that a cubic equation for L can be derived, noting that there is one obvious root.
  • A participant mentions deriving three equations and finding that L can be 1 or 2, but expresses uncertainty about whether there are additional values for L that would also make the vectors dependent.
  • There is a reiteration of the definition of linear dependence, with a question about whether L must be one of the previously identified values for ß to have specific implications.

Areas of Agreement / Disagreement

Participants express uncertainty about the completeness of the values for L that make the vectors dependent. There is no consensus on whether additional values exist beyond those identified.

Contextual Notes

Participants have not fully resolved the implications of the equations derived for L, and there are assumptions regarding the relationship between L and ß that remain unexamined.

gotmejerry
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So i have 3 vectors:
a= [1 1 1]
b= [2 L 0]
c= [L 2 3]

How do I calculate the L in order to make these vecotrs linearly dependent?

How does ß depend from L if v= [ß 0 -1] and v is in span(a b c)?

Thank you!
 
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Do you know what "linearly dependent" means? Use the definition of "linearly dependent" You will get a cubic equation for L but, fortunately, there's 1 obvious root.

For the second question, are we to assume that L is one of those values? Otherwise, the span of a, b, and c is all of R3 and [itex]\beta[/itex] can be anything.
 
I guess I know.

I wrote 3 equations:

α + 2β + Lγ = 0
α + Lβ + 2γ = 0
α + + 3γ = 0

And i got, L can be 1 or 2. Then I checked it and for these Ls the vectors are dependents. But how do I know that there aren't more Ls.

For the second question. Yes L is from those values.
 
HallsofIvy said:
Do you know what "linearly dependent" means? Use the definition of "linearly dependent" You will get a cubic equation for L but, fortunately, there's 1 obvious root.

For the second question, are we to assume that L is one of those values? Otherwise, the span of a, b, and c is all of R3 and [itex]\beta[/itex] can be anything.

But I maybe got the definition wrong.
 

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